Yang–Mills for Mathematicians Infosys-ICTS Ramanujan Lectures: Lecture 1

Yang–Mills for mathematicians Infosys-ICTS Ramanujan Lectures: Lecture 1

Sourav Chatterjee

Sourav Chatterjee Yang–Mills for mathematicians What is this talk about?

I I am going to give you a heavily compressed exposition of a very long story, and four open problems.

I There will be reading references at the end, if you want to seriously learn about it.

I Physicists are generally familiar with most of what I’m going to say, but mathematicians are not. This talk is for mathematicians.

Sourav Chatterjee Yang–Mills for mathematicians Quantum ﬁeld theories

I Quantum ﬁeld theories explain interactions between elementary particles and make predictions about their behaviors.

I Encapsulated by the Standard Model.

I Yang–Mills theories are certain kinds of important quantum ﬁeld theories that constitute the standard model.

I What is a QFT? — This is an open question, not only in mathematics, but also in physics.

I Remarkably, physicists can calculate and make surprisingly accurate predictions using QFTs, without really understanding what these objects are!

I The mathematical construction of quantum ﬁeld theories — more speciﬁcally Yang–Mills theories — is one of the seven millennium problems posed by the Clay Institute.

Sourav Chatterjee Yang–Mills for mathematicians QFT basics

4 I Spacetime: R . Restricted Lorentz transforms: A group of 4 linear transformations of R . I Poincarégroup P consists of all (a, Λ), where Λ is a restricted 4 Lorentz transformation and a ∈ R . 4 I P acts on R as (a, Λ)x = a + Λx. I Special relativity: The laws of physics remain invariant under change of coordinates by the action of the Poincarégroup. I A quantum field theory models the behavior of a physical system (e.g. a collection of elementary particles) using:

I a Hilbert space H, and I a (projective) unitary representation U of P in H. I Assumptions: I To an observer, the state of the system appears as some vector ψ ∈ H. If ψ is known, we can compute probabilities of events. I To a diﬀerent observer, who is using a coordinate system obtained by the action of (a, Λ) on the coordinate system of the ﬁrst observer, the state appears as U(a, Λ)ψ.

Sourav Chatterjee Yang–Mills for mathematicians Time evolution

I Suppose a stationary observer at spatial location (0, 0, 0) observes the physical system in state ψ at time 0. I After time has evolved by t, the observer is in a new coordinate system obtained by the action of ((−t, 0, 0, 0), Id.) on the original coordinate system. (The spacetime point (t, x, y, z) in the original coordinate system becomes (0, x, y, z) in the new coordinate system.) I Thus, after time t, the system will appear to the observer as being in state U((−t, 0, 0, 0), Id.)ψ. I It can be proved that there is a self-adjoint operator H on H so that for any t, U((−t, 0, 0, 0), Id.)ψ = e−itH ψ. I H is called the Hamiltonian. I If a diﬀerent observer was at (x, y, z) at time 0, and moves with constant velocity v for time t, he will observe the system to be in state U((−t, x, y, z), Λ)ψ, where Λ is a restricted Lorentz transformation depending on v.

Sourav Chatterjee Yang–Mills for mathematicians Quantum ﬁeld

I Suppose we are given H and U. 4 I A quantum ﬁeld ϕ is a hypothetical function on R , which 4 when integrated against a smooth test function on R , yields an operator on H.

I To put it more succinctly, it is an operator-valued distribution.

I The quantum field ϕ related to our physical system is a field that satisfies

ϕ(a + Λx) = U(a, Λ)ϕ(x)U(a, Λ)−1.

I The ﬁeld ϕ is used for calculating probabilities of events and expected values of various observables. In fact, it becomes the central object of interest in the study of the system.

Sourav Chatterjee Yang–Mills for mathematicians Wightman axioms

I The most popular approach to giving a fully rigorous definition of a quantum field theory is via the Wightman axioms. I These axioms are essentially a more precise version of what I described in the previous slides. I They include some additional conditions (such as ‘locality’) that must be satisfied by H, U and ϕ, and some assumptions about the existence and properties of a unique vacuum state Ω ∈ H of our system. (This the lowest eigenstate of H.) I The axioms give the bare minimum conditions required to avoid physical inconsistencies. I It has been possible to construct certain simple QFTs, known as free fields, which satisfy the Wightman axioms. I Free fields describe trivial systems of particles that do not interact with each other. I Unfortunately, no one has been able to rigorously construct a nontrivial (interacting) QFT satisfying the Wightman axioms.

Sourav Chatterjee Yang–Mills for mathematicians Calculations???

I If we cannot even define the theory, how can we calculate? I Physicists get around this problem by doing perturbative expansions around free fields. I That is, they assume that the desired QFT is a ‘small perturbation’ of the free field (which is well-defined), and do a kind of Taylor expansion around it. I The calculations involve Feynman diagrams and renormalization. I However, there is a rigorous theorem due to Haag, which says that the Hilbert space for an interacting theory cannot be the same as the Hilbert space for a non-interacting theory. I So it is not clear how one can justify such a perturbative expansion. In fact, in most cases it is not clear what the new Hilbert space is! I And yet, in many cases, these calculations yield results that match experiments to remarkable degrees of accuracy.

Sourav Chatterjee Yang–Mills for mathematicians The probabilistic approach (Constructive QFT)

I There is a probabilistic approach to constructing QFTs that satisfy the Wightman axioms. It goes as follows: 4 I First, construct a random field ξ on R whose probability law is related to the desired QFT in a certain way. Usually this is a random distribution, and not a random function. I ξ is called a Euclidean QFT. I Show that ξ satisfies a set of conditions known as the Osterwalder–Schrader axioms. I If this is true, then there is a reconstruction theorem that allows us to construct the desired QFT (i.e., H, U, ϕ and Ω.) I In general, the QFT is nontrivial if and only if the field ξ is non-Gaussian. I The program, initiated in the 60s, was successful in constructing nontrivial QFTs when the dimension of spacetime was reduced from 4 to 2 or 3 — but not yet in dimension 4. 4 I The most notable achievements were the constructions of ϕ2 4 and ϕ3 theories (in spacetime dimensions 2 and 3, respectively). Sourav Chatterjee Yang–Mills for mathematicians Yang–Mills theories

4 I ϕ theories are mathematically interesting, but describe no real physical system.

I To venture into the real world, one has to consider 4D Yang–Mills theories.

I These are QFTs that describe interactions between real elementary particles.

I The question is completely settled in 2D. I There was a tremendous amount of work on rigorously constructing Yang–Mills theories in 3D and 4D, by Ba laban and others.

I However, the investigation was inconclusive and the question is still considered to be open.

I Even the ﬁrst step in the probabilistic approach, namely, the construction of a random ﬁeld, remains open. We will now talk about that.

Sourav Chatterjee Yang–Mills for mathematicians Euclidean Yang–Mills theories

I Recall that the ﬁrst step in the probabilistic approach to constructing QFTs is the construction of a suitable random ﬁeld, known as a Euclidean QFT.

I For Yang–Mills theories, these random ﬁelds are called Euclidean Yang–Mills theories.

I These have not yet been constructed in spacetime dimensions 3 and 4.

I Euclidean Yang–Mills theories are supposed to be scaling limits of lattice gauge theories, which are well-deﬁned discrete probabilistic objects, which I will now discuss.

Sourav Chatterjee Yang–Mills for mathematicians Lattice gauge theories

I Let d = dimension of spacetime, and G be a matrix Lie group. (Most important: d = 4 and G = SU(2) or SU(3).) I The lattice gauge theory with gauge group G on a finite set d Λ ⊆ Z is defined as follows. I Suppose that for any two adjacent vertices x, y ∈ Λ, we have a group element U(x, y) ∈ G, with U(y, x) = U(x, y)−1. I Let G(Λ) denote the set of all such configurations. I A square bounded by four edges is called a plaquette. Let P(Λ) denote the set of all plaquettes in Λ. I For a plaquette p ∈ P(Λ) with vertices x1, x2, x3, x4 in anti-clockwise order, and a configuration U ∈ G(Λ), define

Up := U(x1, x2)U(x2, x3)U(x3, x4)U(x4, x1).

I The Wilson action of U is deﬁned as X SW(U) := Re(Tr(I − Up)). p∈P(Λ)

Sourav Chatterjee Yang–Mills for mathematicians Deﬁnition of lattice gauge theory contd.

I Let σΛ be the product Haar measure on G(Λ).

I Given β > 0, let µΛ,β be the probability measure on G(Λ) deﬁned as 1 dµ (U) := e−βSW(U)dσ (U), Λ,β Z Λ where Z is the normalizing constant.

I This probability measure is called the lattice gauge theory on Λ for the gauge group G, with inverse coupling strength β.

I An inﬁnite volume limit of the theory is a weak limit of the d above probability measures as Λ ↑ Z (may not be unique).

Sourav Chatterjee Yang–Mills for mathematicians Open problem #1: Yang–Mills existence

I To define the scaling limit of a lattice gauge theory, one has to d first define it on the scaled lattice Z and then send → 0. I To obtain an interesting limit, one has to vary the parameter β as → 0. I In dimension 3, it is believed β has to scale like a multiple of −1, and in dimension 4, it is believed that β has to scale like a multiple of log(1/). I The most interesting gauge groups are non-Abelian Lie groups like SU(2) and SU(3). I It is not clear what the scaling limit should look like, or what space it should belong to. I Even if one is able to somehow obtain a scaling limit, it is important to prove that it is nontrivial — meaning that it is a non-Gaussian field (on whatever space it’s defined on). I Finally, one has to construct the actual QFT using this field, via the Osterwalder–Schrader axioms or otherwise.

Sourav Chatterjee Yang–Mills for mathematicians YM existence: Mathematical literature

I Well-understood in dimension 2. Many contributors.

I In dimensions 3 and 4, long series of papers by Balaban in the 80s, aiming to prove the existence of subsequential scaling limits. Established results about the behavior of the partition function in the scaling limit.

I Another attempt in 1994 by Magnen, Rivasseau and S´en´eor.

I However, the problem is still considered to be open in dimensions 3 and 4.

I Hairer’s theory of regularity structures recently allowed a 4 diﬀerent construction of ϕ3 theory via stochastic quantization. Ongoing work by Hairer and collaborators to extend this approach to 3D Yang–Mills theories.

Sourav Chatterjee Yang–Mills for mathematicians Open problem #2: Mass gap

I Recall the Hamiltonian H of a QFT, and the vacuum state Ω. The vacuum state is the unique (up to scalar multiples) nonzero element of H that satisﬁes HΩ = 0.

I The theory is said to have a mass gap if there is some µ > 0 such that any other eigenvalue of H is ≥ µ.

I Physically, this means that the particles described by the theory possess nonzero mass.

I If we go through the probabilistic approach, the mass gap question can be shown to be equivalent to the question of exponential decay of correlations in the Euclidean QFT.

I Various Yang–Mills theories — such as 4D Yang–Mills theory with gauge group SU(3) — are supposed to have mass gaps.

I The ﬁrst step to showing this is to show that the corresponding lattice gauge theories have exponential decay of correlations at large β.

Sourav Chatterjee Yang–Mills for mathematicians Mass gap: Mathematical literature

I At small β, exponential decay can be proved by well-known techniques from statistical physics (expansions or coupling).

I No general method for large β.

I There are a number of simpler models where exponential decay is conjectured at large β but not proved. Examples: 2D Ising type models with vector spins (the 2D O(N) models with N ≥ 3).

I A rare example where this has been proved recently: The loop O(N) model, a toy version of the O(N) model, by Duminil-Copin et al.

Sourav Chatterjee Yang–Mills for mathematicians Wilson loops

d I Consider a lattice gauge theory on Z with gauge group G. I Let U be a random conﬁguration of group elements attached to edges, drawn from the probability measure deﬁned by this theory.

I Given a loop γ with directed edges e1,..., em, the Wilson loop variable Wγ is deﬁned as

Wγ := Re(Tr(U(e1)U(e2) ··· U(em))).

I The expected value of Wγ is denoted by hWγi.

Sourav Chatterjee Yang–Mills for mathematicians Quark conﬁnement

I Lattice gauge theories and Wilson loops were introduced by Wilson in 1974 primarily to understand the phenomenon of quark conﬁnement.

I Quarks are elementary particles that bind together to form protons, neutrons, etc.

I Quarks are always bound, and never occur freely in nature. This is known as quark conﬁnement or color conﬁnement.

I Wilson argued that this phenomenon occurs due to a mathematical feature of Yang–Mills theories, that is now called Wilson’s area law.

Sourav Chatterjee Yang–Mills for mathematicians Open problem #3: Quark conﬁnement

I Take any 4D non-Abelian lattice gauge theory.

I Show that for any β, there are constants C(β) and c(β) such that for any loop γ,

−c(β)area(γ) |hWγi| ≤ C(β)e ,

where hWγi is the expected value of the Wilson loop variable Wγ and area(γ) is the minimal surface area enclosed by γ. I This is known as Wilson’s area law, and was argued by Wilson to be the reason behind conﬁnement of quarks.

I Showing for rectangles is good enough.

Sourav Chatterjee Yang–Mills for mathematicians Quark conﬁnement: Mathematical literature

I There is a general proof at small β by Osterwalder and Seiler (1978).

I Proof at large β for 3D U(1) theory by G¨opfertand Mack (1982).

I Disproof at large β for 4D U(1) theory by Guth (1980) and Fr¨ohlichand Spencer (1982). Therefore in 4D at large β, it is crucial that the gauge group is non-Abelian.

Sourav Chatterjee Yang–Mills for mathematicians Gauge-string duality

I Unifying quantum mechanics and gravity is the holy grail of modern physics.

I In 1997, Maldacena made the remarkable discovery that certain quantum ﬁeld theories are ‘dual’ to certain string theories.

I Duality means that any calculation in one theory corresponds to some calculation in the other theory.

I Maldacena’s discovery is known as AdS-CFT duality or gauge-string duality or gauge-gravity duality.

Sourav Chatterjee Yang–Mills for mathematicians Open problem #4: Gauge-string duality

I To establish gauge-string duality for YM theories, one can, for example, try to show that expected values of Wilson loop variables are expressible as integrals over trajectories of strings in a string theory.

I Tremendous activity in physics, but almost nothing on the mathematical side. Possibly because the relevant QFTs are not mathematically well-deﬁned.

I In 2015, I proved such a result for lattice gauge theories at small β — probably the ﬁrst mathematical theorem in this area. This will be the topic of the next talk.

I However, this is a discrete result. It is an open problem to prove such a theorem when β is large. We need to consider large β for passing to the continuum limit.

Sourav Chatterjee Yang–Mills for mathematicians Where to read about all this

I My preprint “Yang–Mills for probabilists” on arXiv has more details for many of the topics presented here.

I On my website, you will ﬁnd lecture notes for a course on “Quantum ﬁeld theory for mathematicians” that I taught recently at Stanford. Introduces the foundations of QFT but not Yang–Mills.

I The above lecture notes are based on a (terriﬁc) forthcoming book by Michel Talagrand that presents QFT for a mathematical audience.

I Hairer’s stochastic quantization approach is explained in various surveys and lecture notes, besides the original papers.

I Constructive QFT is explained in the textbook of Glimm and Jaﬀe, and in many surveys and expositions available online.

I Further references in the next talk.

Sourav Chatterjee Yang–Mills for mathematicians